Team Review

• How did everyone feel about the problem set?

• Any questions?

• Discuss with study partner

Functions are like baking recipes

What is a function?

For each combination of independent variables, there is exactly one (and only one) value of the dependent variable.

Example

A function \(f(x)\) cannot return two values for \(x=1\): \(f(1)\ne\{2,3\}\)

Vertical line test

If you run a vertical line perpendicular to the \(x\)-axis and see where it intersects the graph of a function, then it should only intersect it once at every value of \(x\).

Continuity

Intuitively, a function is continuous if there are no ‘breaks’ in the graph of the function. This means you are able to draw the whole graph without lifting your pencil.

To define it formally we need to understnad limits.

A function is discontinuous if it’s not continuous! So there are ‘breaks’ or ‘gaps’ in its graph.

Limits

Definition

We say a number \(L\) is the limit of a function \(f(x)\) as its variable \(x\) approaches a number \(c\), if the function’s output values \(f(x)\) approach \(L\) when \(x\) get closer to \(c\).

We write this symbolically as:

\[ \large \lim_{x\to c} f(x)=L \]

We read this as: “The limit of \(f(x)\) as \(x\) approaches \(c\) is \(L\).”

Example: numerical

What is the limit of \(f(x)=2x^2-4\) as \(x\) approaches 2?

We need to examine the values of \(f(x)\) as \(x\) gets closer to 2 from both sides.

Example: graph

What is the limit of \(f(x)=2x^2-4\) as \(x\) approaches 2?

From both directions it looks like \(f(x)\) converges to 4.

So, \(\lim_{x\to 2}(2x^2-4)=4\).

Continuity revisited

What happens with discontinuous functions?

Try with your group the same approach for:

\[ \lim_{x\to 2}f(x)=\frac{|x-2|}{x-2} \]

Hint: Start very close to 2 like 1.9 and 2.1

We broke the universe

Clearer on a graph

Technically the limit is “Undefined”, because it does approach a finite value. Does not exist happens when the limits spiral off to infinity in both directions (tan graph)

Defined Limit at an undefined point

• Discontinuous functions can still have limits

• Find the limit of \(f(x)=x+1,x\ne 2\) as x approaches 2

Work with your team to complete these task

Work with your team to complete these task

2. Can you think of examples where discontinuous functions might exist in environmental science?

3. Choose as a team to draw an example of one of these statements:

• \(\lim_{x\to 4^-} f(x)\) and \(\lim_{x\to 4^+}f(x)\) are both infinite

• \(\lim_{x\to 3} f(x)=2\), but \(f(3)=0\)

• \(\lim_{x\to 5^-} f(x)=4\) and \(\lim_{x\to 5^+} f(x)=2\)

• \(\lim_{x\to -3} f(x)=-5\) but \(f(-3)=-5\)

Putting it all together

Recall average rate of change and instantaneous

Taking the average rate of change to a set limit will eventually converge to the instantaneous.

Walk Through Example

Walk Through Example

Walk Through Example

Walk Through Example

Walk Through Example

Walkthrough

Tangent Lines

What if we set \(\Delta x=0\)? Then we would have a slope line that only touches our function at exactly \(x\).

These are called tangent lines

Put our example in math notation

\[\text{slope}=\frac{y_2-y_1}{x_2-x_1}\]

Choose a point along the function \((x,f(x))\)

Choose a different point on the function \(\Delta x\) away \((x+\Delta x,f(x+\Delta x))\)

Add these into the slope equation

\[ \text{slope}=\frac{f(x+\Delta x)-f(x)}{(x+\Delta x) -x}=\frac{f(x+\Delta x)-f(x)}{\Delta x} \]

Derivative Definition

\[ \large f'(x)=\lim_{\Delta x \to 0}=\frac{f(x+\Delta x)-f(x)}{\Delta x} \]

\[ \large f'(x)\text{, or }\frac{dy}{dx} \]

Not all functions are differentiable

Calculus and Derivatives are the study of change

Environmental Science is also a study of change

Environmental Science is also a study of change

Environmental Science is also a study of change

Rules for Differentiation

• If \(f(x)\) is contant, then for all \(x\), \(f'(x)=0\)

\[ \begin{align} &y=a &\frac{dy}{dx}=0 \end{align} \]

\[ \frac{d}{dx}[x^n] =nx^{n-1} \]

Examples

\[ \begin{align} &y=100 & &y=5x^5 & &y=\frac{1}{x^2} \end{align} \]

Rules for Differentiation

\[ \begin{align} \frac{d}{dx}&=[f(x)+g(x)]=f'(x)+g'(x) \\ \frac{d}{dx}&=[f(x)-g(x)]=f'(x)-g'(x) \end{align} \]

Looks really scary. All it says, if the function has pieces that are added or subtracted you can take the derivative of each individual piece.

Example

\[ \begin{align} &y=x^2+8x+4 & &y=x^3-x^2+x-15 \end{align} \]

1. As a team, list 5 fields of environmental science where studying the rate of change and derivatives would be important

2. Which rules should you use to take these derivatives?

\[ \begin{align} \text{A) }& f(x)=3x^4 &\text{B) } y=4x^2+3x-16 \end{align} \]

3. Find the derivatives of these functions

\[ \begin{align} &\text{A) } y=3x^2 & &\text{B) }h(x)=y=7x+4 & &\text{C) }g(y)=\sqrt{y} \end{align} \]

4. Discuss why derivatives need continuity. It might help to think about why some continuous functions don’t have derivatives.